Witte & Witte, 9e Page 1 of 6 Pages
Chapter 20
Chapter 20: Tests for Ranked (Ordinal) Data
Exercise 1
A researcher wanted to find out if there was difference between older movie goers and younger movie goers with respect to their estimates of a successful actor’s income. The researcher first identified Tom Hanks as the top money maker based on data provided at www.boxofficemojo.com. The researcher then asked a random sample of older movie goers (50-60 years) and a random sample of younger movie goers (20-30 years) to give their best estimate of Tom Hanks’ average earnings per film. (Tom Hanks’ average, according to www.boxoffice.mojo.com, is $100.9 million). Carry out a nondirectional Mann-Whitney U Test on the estimates given by the two samples to find out if the distributions are equal. Follow the steps given below the table and use a significance level of .05.
Estimate of Tom Hanks'Average Earnings Per Film
(in millions)
Older Movie Goers / Younger Movie Goers
$40 / $80
$50 / $100
$200 / $90
$8 / $50
$2 / $1
$10 / $6
$12 / $15
$15 / $60
$20 / $300
$50 / $40
a. Present the null hypothesis.
b. Present the alternative hypothesis.
c. Identify the critical value of U by referring to Table E in your textbook.
d. Present the decision rule.
e. Calculate U.
f. Present a statistical decision.
g. Present a conclusion in the context of the research situation.
Answers:
a. H0: Population distributionOlder = Population distributionYounger
b. H1: Population distributionOlder ≠ Population distributionYounger
c. Critical U = 23
d. Reject H0 at the .05 level of significance if U ≤ 23.
RanksOlder Movie Goers / Younger Movie Goers
10.5 / 16
13 / 18
19 / 17
4 / 13
2 / 1
5 / 3
6 / 7.5
7.5 / 15
9 / 20
13 / 10.5
89 / 121
= 61
= 29
U = 29 because it is the smaller of U1 and U2.
f. Reject H0.
g. The distributions are not equal. An inspection of the data indicates that the younger movie goers’ estimates are higher than those of the older movie goers.
Exercise 2
A researcher wanted to find out if there was difference between the estimates given for a female actor’s earnings and a male actor’s earnings. The researcher hypothesized that the estimates would be higher for a male actor than for a female actor. The researcher first identified Tom Hanks as the top male money maker and Julia Roberts as the top female money maker based on data provided at www.boxofficemojo.com. The researcher then asked a random sample of movie goers to give their best estimates of Tom Hanks’ average earnings per film and Julia Roberts’ average earnings per film. (Julia Roberts’ average, according to www.boxoffice.mojo.com, is $70.1 million). Carry out a directional Wilcoxon T Test for two related samples on the estimates to find out if the distributions are equal. Follow the steps given below the table and use a significance level of .05 for the directional test.
Estimate of Actors' Average Earnings Per Film(in millions)
Tom Hanks / Julia Roberts
$1 / $1
$10 / $10
$15 / $80
$25 / $100
$200 / $10
$5 / $20
$100 / $15
$10 / $60
a. Present the null hypothesis.
b. Present the alternative hypothesis.
c. Determine n, the number of nonzero difference scores.
d. Identify the critical value of T in Table F of your textbook.
e. Present the decision rule.
f. Calculate T.
g. Present a statistical decision.
h. Present a conclusion in the context of the research situation. You may assume that both populations have the same variability and the same shape.
Answers:
a. H0: Population distributionTom Hanks = Population distributionJulia Roberts
b. H1: Population distributionTom Hanks ≠ Population distributionJulia Roberts
c. n = 6
Estimate of Actors' Average Earnings Per Film(in millions)
Tom Hanks / Julia Roberts / Difference Score
$1 / $1 / $0
$10 / $10 / $0
$15 / $80 / -$65
$25 / $100 / -$75
$200 / $10 / $190
$5 / $20 / -$15
$100 / $15 / $85
$10 / $60 / -$50
d. Critical value of T = 2
e. Reject H0 at the .05 level if T ≤ 2.
f. T = 10 because it is smaller than 11.
Estimate of Actors' Average Earnings Per Film(in millions)
Tom Hanks / Julia Roberts / Difference Score / Ordered Scores / Ranks / Plus Ranks / Minus Ranks
$1 / $1 / $0 / -$15 / 1 / 1
$10 / $10 / $0 / -$50 / 2 / 2
$15 / $80 / -$65 / -$65 / 3 / 3
$25 / $100 / -$75 / -$75 / 4 / 4
$200 / $10 / $190 / $85 / 5 / 5
$5 / $20 / -$15 / $190 / 6 / 6
$100 / $15 / $85 / 11 / 10
$10 / $60 / -$50
g. Retain H0 at the .05 level because T = 10 is not less than 2.
h. There is no evidence that, on average, estimates of the earnings of a successful male actor exceed that for a successful female actor.
Exercise 3
A researcher wants to compare the salaries of registered nurses who work in California, New York, and Florida. The researcher randomly selected several RN’s in each state and records their annual salaries. The salaries are listed in the table shown below. Follow these steps, using a = .05, to see if you can conclude that the distributions in these three states are different.
Annual SalariesCalifornia / New York / Florida
$40,500 / $43,400 / $39,250
$56,060 / $45,800 / $47,620
$42,350 / $55,650 / $48,500
$74,200 / $74,200 / $62,400
$65,200 / $33,480 / $58,200
$43,400 / $55,800 / $32,100
$41,900 / $49,200 / $34,570
$68,500
a. Present the null hypothesis.
b. Present the alternative hypothesis.
c. Calculate df.
d. Identify the critical value of H in Table D of your textbook.
e. Present the decision rule.
f. Calculate H.
g. Present a statistical decision.
h. Present a conclusion in the context of the research situation.
Answers:
a. H0: PopulationCalifornia = PopulationNew York = PopulationFlorida
b. H1: H0 is false
c. df = number of groups – 1 = 3 – 1 = 2
d. Critical H = 5.99
e. Reject H0 at the .05 level of significance if H ≥ 5.99.
f.
Annual SalariesCalifornia / New York / Florida
5 / 8.5 / 4
16 / 10 / 11
7 / 14 / 12
21.5 / 21.5 / 18
19 / 2 / 17
8.5 / 15 / 1
6 / 13 / 3
20
103 / 84 / 66
= = 1.11
g. Retain H0 at the .05 level of significance because H = 1.11 is less than 5.99.
h. There is no evidence that RN salaries differ in California, New York, and Florida.
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